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arxiv: 2605.29069 · v1 · pith:EEFJRSH2new · submitted 2026-05-27 · 🌌 astro-ph.GA · astro-ph.CO· gr-qc· math-ph· math.MP

Equilibrium Core and Vortex Solutions of Bose Einstein Condensate Dark Matter around a Black Hole

Ivan Alvarez-Rios , Francisco S. Guzman This is my paper

Pith reviewed 2026-06-29 10:28 UTC · model grok-4.3

classification 🌌 astro-ph.GA astro-ph.COgr-qcmath-phmath.MP
keywords Bose-Einstein condensate dark matterblack holevortex solutionsaxisymmetric equilibriastability analysisself-interactionenthalpy functional
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The pith

BEC dark matter forms stable cores and vortices around a black hole

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs families of stationary axisymmetric solutions for Bose-Einstein condensate dark matter surrounding a central black hole, covering both ground-state cores and first-winding-number vortex lines. An imaginary-time method solves the coupled equations across ranges of self-interaction strength and black hole mass, yielding density profiles that vary systematically with those parameters. Stability along each family is classified by the turning point criterion applied to the enthalpy functional, separating stable from unstable branches. For attractive self-interaction the solutions are bounded by a maximum mass, which restricts the parameter window in which the configurations can form. This extends earlier results on core solutions acting as attractors in collapse to show that vortex solutions can likewise occupy stable branches compatible with formation scenarios.

Core claim

Stationary axisymmetric solutions of BECDM around a point-mass black hole exist for both the ground-state core and the line vortex with nonzero winding number; families of these solutions are obtained for varying self-interaction and black hole mass, with the turning point criterion on the enthalpy functional identifying stable branches, and a maximum mass appearing when the self-interaction is attractive.

What carries the argument

Imaginary-time relaxation of the axisymmetric stationary Gross-Pitaevskii-Poisson system, combined with the turning point criterion on the enthalpy functional to separate stable and unstable branches.

If this is right

  • Core solutions continue to act as attractors in collapse around a black hole.
  • Vortex solutions with nonzero winding number possess stable branches.
  • Attractive self-interaction imposes a maximum mass that bounds the allowable parameter range.
  • Density profiles change continuously with self-interaction strength and black hole mass.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the stable vortex branches survive in full dynamical simulations they could influence angular-momentum transport near galactic-center black holes.
  • The same numerical construction could be repeated for Kerr black holes to test whether frame-dragging alters the location of the stability turning points.
  • The maximum-mass bound for attractive interactions supplies a concrete target for N-body or fluid simulations of BECDM halo assembly around supermassive black holes.

Load-bearing premise

The turning point criterion applied to the enthalpy functional correctly classifies stable and unstable branches for these axisymmetric, self-gravitating BEC configurations.

What would settle it

A long-term numerical time evolution in which a configuration labeled stable by the turning-point criterion remains intact while one labeled unstable collapses or disperses would confirm or refute the stability classification.

Figures

Figures reproduced from arXiv: 2605.29069 by Francisco S. Guzman, Ivan Alvarez-Rios.

Figure 1
Figure 1. Figure 1: FIG. 1. Radial profiles on the equatorial plane ( [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Invariant mass [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Maximum invariant mass ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Critical invariant ( [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

We present the construction of stationary solutions of Bose-Einstein condensate dark matter (BECDM) around a point-like gravitational source representing a black hole. The problem is formulated for general axisymmetric configurations, and we focus on two cases: the ground-state core solution and the first nonzero winding number configuration corresponding to a line vortex solution. The stationary equations are solved using an imaginary-time approach, which enables the construction of families of solutions across a wide range of self-interaction and black hole masses. We analyze the impact of these parameters on the density distribution and on the stability properties of the solutions, assessing stability through the turning point criterion based on the enthalpy functional, which allows us to identify stable and unstable branches along each family of solutions. It has been shown in the past that spherical core solutions act as attractors in the collapse of BECDM around black holes in the non-interacting case ($g=0$), supporting their astrophysical relevance. In the present work, the existence of a maximum mass for configurations with attractive self-interaction ($g<0$) allows us to infer the parameter range in which such solutions may also arise in this regime. Building on this picture, we show that stable vortex solutions of BECDM can also exist in the presence of a black hole, whose stability properties suggest that these configurations may likewise be compatible with physically relevant formation scenarios.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript constructs stationary axisymmetric solutions of the Gross-Pitaevskii-Poisson system describing BECDM around a central point-mass black hole. It obtains families of ground-state core solutions and first-excited vortex solutions (nonzero winding number) via an imaginary-time method across ranges of self-interaction strength g and black-hole mass. Stability is classified by applying the turning-point criterion to the enthalpy functional, identifying stable and unstable branches; for attractive g<0 a maximum mass is reported, and the existence of stable vortices is argued to be compatible with astrophysical formation scenarios.

Significance. If the stability classification holds, the work extends earlier results on spherical g=0 cores to include vortices and nonzero g, supplying concrete parameter ranges (maximum mass for attractive interactions) that could be tested against formation simulations. The systematic construction of solution families parameterized by g and M_BH is a methodological strength.

major comments (1)
  1. [Abstract and stability analysis] Abstract (final paragraph) and stability analysis: the turning-point criterion on the enthalpy functional is used to classify stable versus unstable vortex branches, yet no derivation or cross-validation is supplied showing that the one-parameter turning-point argument remains equivalent to dynamical stability once nonzero phase winding and the central point-mass singularity are present. This directly underpins the central claim that stable vortex solutions exist and are astrophysically relevant.
minor comments (2)
  1. [Abstract] The abstract would benefit from a concise statement of the governing equations and the explicit definition of the enthalpy functional used for the turning-point test.
  2. [Numerical method] Convergence tests, grid resolution, and residual norms for the imaginary-time solver should be reported to allow assessment of the accuracy of the density profiles and masses entering the stability analysis.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. Below we respond point-by-point to the single major comment, indicating the revision that will be made to address the concern about the turning-point criterion.

read point-by-point responses

  1. Referee: [Abstract and stability analysis] Abstract (final paragraph) and stability analysis: the turning-point criterion on the enthalpy functional is used to classify stable versus unstable vortex branches, yet no derivation or cross-validation is supplied showing that the one-parameter turning-point argument remains equivalent to dynamical stability once nonzero phase winding and the central point-mass singularity are present. This directly underpins the central claim that stable vortex solutions exist and are astrophysically relevant.

    Authors: We acknowledge that the manuscript applies the turning-point criterion to the enthalpy functional without supplying an explicit derivation or numerical cross-validation tailored to the combination of nonzero winding number and central point-mass singularity. The criterion follows from the variational structure of the GPP energy functional (extremization with respect to density at fixed particle number), which has been used previously for spherical cores and for vortex states in the absence of a central singularity. The addition of a fixed central potential and azimuthal phase winding enters the effective potential but preserves the one-parameter family structure used in the turning-point analysis. Nevertheless, we agree that an explicit statement of applicability strengthens the central claim. In the revised manuscript we will add a concise paragraph (or short appendix) in the stability section that (i) recalls the standard derivation of the turning-point method for the GPP system and (ii) notes why the same argument carries over to axisymmetric configurations with winding and an external 1/r potential. This revision will be made without altering the reported solution families or stability classifications. revision: yes

Circularity Check

0 steps flagged

No significant circularity; numerical construction from GPP system

full rationale

The paper numerically constructs stationary axisymmetric solutions (core and vortex) of the Gross-Pitaevskii-Poisson system with central point-mass potential via imaginary-time relaxation, then applies the standard turning-point criterion to the enthalpy functional along families parameterized by g and black-hole mass. No quoted step reduces a claimed prediction or stability classification to a fitted input, self-defined quantity, or load-bearing self-citation chain; the attractor statement for g=0 cores is external prior support. The derivation remains self-contained against the underlying equations.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Paper rests on standard numerical assumptions for BECDM; no new entities or heavily fitted parameters are introduced in the abstract.

free parameters (2)
  • self-interaction strength g
    Varied parametrically to generate solution families; not fitted to external data.
  • black hole mass
    Varied parametrically to generate solution families; not fitted to external data.
axioms (2)
  • domain assumption Imaginary-time evolution converges to stationary solutions of the axisymmetric Gross-Pitaevskii equation with Newtonian gravity.
    Invoked to construct the families.
  • domain assumption Turning-point criterion on the enthalpy functional identifies dynamical stability for these configurations.
    Used to label stable and unstable branches.

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    Assuming spherical symmetry of the total potential, 11 VT (r⊥, z) =V T (r) withr= p r2 ⊥ +z 2, the ground state inherits this symmetry so thatϕ 0(r⊥, z) =ϕ(r). Using the chain rule,∂ r⊥ ϕ0 =ϕ ′(r)r ⊥/rand∂ zϕ0 = ϕ′(r)z/r, the regularity conditions are automatically sat- isfied. The Laplacian becomesϕ ′′(r) + 2 r ϕ′(r), and the equation reduces to −1 2 ϕ′′...

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    Regularity of the Axialm >0Vortex Solution For a vortex configuration with winding numberm > 0, the axial stationary equation is −1 2 ∂2ϕm ∂r2 ⊥ + 1 r⊥ ∂ϕm ∂r⊥ − m2 r2 ⊥ ϕm +∂ 2 z ϕm + VT (r⊥, z)ϕm =µϕ m. Near the symmetry axis, the dominant contribution is the centrifugal term−m 2ϕm/r2 ⊥, so to leading order the equation reduces to∂ 2 r⊥ ϕm + 1 r⊥ ∂r⊥ ϕm...